In the tutorial I could pick up 2 potions and store them in my inventory. But I can't do the same in the "real" game where, upon pressing the use key, the character will drink (or won't do anything, if HP are full) the potion automatically. Do I need to find a bag or something ?

edit : nevermind, looks like some potions just cannot be stored, like the life 150 potion.
Well so that post won't be here for nothing, here's another question : are there any destructible walls or hidden passages in this game ? (I mean, not obvious ones like a wall full of cracks for example)

potion levels 50-200 are automatic. You'll notice in your inventory, potions of powers of 500-Full will go there.

I think most (if not all) passages/secrets are only a little off the beaten track, and lock or switch related, though I recall more than one collapsing floor.